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  "body": "\n## Definition\n\nLet $d \\ge 1$ and $n \\ge d + 1$, and let $v_1, \\dots, v_n$ be points of $\\R^d$ whose [[affine span|0YMB1676R1]] is $\\R^d$.\nLet $A$ be the $(d+1) \\times n$ matrix whose $i$-th column is $(v_i, 1) \\in \\R^{d+1}$.\nIts [[null space|0PHDYNR24D]] $N = \\set{\\lambda \\in \\R^n : A \\lambda = 0}$, the space of affine dependences of the points, has dimension $n - d - 1$.\nChoose a basis of $N$ and let $B$ be the $n \\times (n - d - 1)$ matrix with these basis vectors as columns.\nThe *Gale transform* of $v_1, \\dots, v_n$ is the sequence $g_1, \\dots, g_n \\in \\R^{n-d-1}$ of rows of $B$.\n\n## Remarks\n\nThe Gale transform is determined up to an invertible linear map of $\\R^{n-d-1}$, the change of basis of $N$.\nMany authors call the configuration $g_1, \\dots, g_n$ a *Gale diagram*, and some first rescale each nonzero $g_i$ by a positive factor; this does not affect the property below.\n\nKey property: let $T \\subseteq \\set{1, \\dots, n}$ be nonempty and $S$ its complement.\nConsider an affine function $\\ell$ on $\\R^d$ with $\\ell(v_i) = 0$ for $i \\in S$ and $\\ell(v_i) \u003e 0$ for $i \\in T$.\nSuch an $\\ell$ exists if and only if $0$ lies in the relative interior of the [[convex hull|0FQSDC2EQ2]] of $\\set{g_i : i \\in T}$.\nIndeed, the vectors $(\\ell(v_1), \\dots, \\ell(v_n))$ are exactly the vectors $y$ with $\\sum_i y_i g_i = 0$.\nSo when the $v_i$ are the vertices of a [[convex polytope|300K6CB1KH]] $P$, the sets $\\set{v_i : i \\in S}$ that are vertex sets of faces of $P$ are read off from the $g_i$.\n\nExample: for the $4$ vertices $0, e_1, e_2, e_1 + e_2$ of a square in $\\R^2$, $n - d - 1 = 1$, and $N$ is spanned by $(1, -1, -1, 1)$.\nSo the Gale transform is $1, -1, -1, 1$ in $\\R$.\nFor $T = \\set{1, 4}$, the hull of $g_1 = g_4 = 1$ is $\\set{1}$, which misses $0$, so $\\set{e_1, e_2}$ is not a face: it is a diagonal.\nFor $T = \\set{1, 2}$, the hull of $1$ and $-1$ contains $0$ in its interior, so $\\set{e_2, e_1 + e_2}$ is an edge.\n\nNon-example: the points $0, e_1, 2e_1$ in $\\R^2$ do not span $\\R^2$ affinely, so the construction does not apply.\n",
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